INTERNATIONAL
GCSE
Further Pure Mathematics (9-1)
SPECIFICATION
Pearson Edexcel International GCSE in Further Pure Mathematics (4PM1)
For first teaching September 2017
First examination June 2019
INTERNATIONAL GCSE
Further Pure Mathematics
SPECIFICATION
Pearson Edexcel International GCSE in Further Pure Mathematics
(4PM1)
For first teaching in September 2017
First examination June 2019
Edexcel, BTEC and LCCI qualifications
Edexcel, BTEC and LCCI qualifications are awarded by Pearson, the UK’s largest awarding
body offering academic and vocational qualifications that are globally recognised and
benchmarked. For further information, please visit our qualification websites at
qualifications.pearson.com. Alternatively, you can get in touch with us using the details on
our contact us page at qualifications.pearson.com/contactus
About Pearson
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countries working to help people of all ages to make measurable progress in their lives
through learning. We put the learner at the centre of everything we do, because wherever
learning flourishes, so do people. Find out more about how we can help you and your
learners at qualifications.pearson.com
Acknowledgements
Pearson has produced this specification on the basis of consultation with teachers,
examiners, consultants and other interested parties. Pearson would like to thank all those
who contributed their time and expertise to the specification’s development.
References to third party material made in this specification are made in good faith. Pearson
does not endorse, approve or accept responsibility for the content of materials, which may
be subject to change, or any opinions expressed therein. (Material may include textbooks,
journals, magazines and other publications and websites.)
All information in this specification is correct at time of going to publication.
ISBN 978 1 446 93113 4
All the material in this publication is copyright
© Pearson Education Limited 2016
Contents
1
About this specification
1
Using this specification
1
Qualification aims and objectives
1
Why choose Edexcel qualifications?
2
Specification updates
1
Why choose Pearson Edexcel International GCSE in Further Pure
Mathematics?
2
Supporting you in planning and implementing this qualification
3
Qualification at a glance
5
Paper overview
5
2
Further Pure Mathematics content
7
3
Assessment information
4
21
Assessment requirements
21
Calculators
22
Assessment objectives and weightings
23
Relationship of assessment objectives to units
23
Administration and general information
25
Entries
25
Access arrangements, reasonable adjustments, special
consideration and malpractice
25
Language of assessment
25
Access arrangements
25
Reasonable adjustments
26
Special consideration
26
Further information
26
Malpractice
26
Candidate malpractice
26
Staff/centre malpractice
27
Awarding and reporting
27
Student recruitment and progression
27
Prior learning and other requirements
27
Progression
27
Appendices
29
Appendix 1: Codes
31
Appendix 2: Pearson World Class Qualification Design Principles
33
Appendix 3: Transferable skills
35
Appendix 4: Formulae sheet for examinations
37
Appendix 5: Formulae to learn
39
Appendix 6: Notation
43
Appendix 7: Glossary
45
1 About this specification
The Pearson Edexcel International GCSE in Further Pure Mathematics is part of a suite of
International GCSE qualifications offered by Pearson.
This qualification is not accredited or regulated by any UK regulatory body.
This specification includes the following key features.
Structure: the Pearson Edexcel International GCSE in Further Pure Mathematics is a linear
qualification. It consists of two examinations available at Higher Tier only
(targeted at grades 9–4, with 3 allowed). Both examinations must be taken in the same
series at the end of the course of study.
Content: relevant, engaging, emphasising the importance of further pure mathematics at
International GCSE Level. Constructed to extend knowledge of the further pure mathematics
topics in the specifications for the Pearson Edexcel International GCSE in Mathematics
(Specification A) (Higher Tier) and the Pearson Edexcel International GCSE in Mathematics
(Specification B).
Assessment: designed to be accessible to students from grades 9–4 using varying
question-style approaches and the introduction of a formulae sheet in the written
examinations.
Approach: a solid basis for students wishing to progress to Edexcel AS and Advanced
GCE Level, or equivalent qualifications.
Specification updates
This specification is Issue 1 and is valid for the Pearson Edexcel International GCSE in
Further Pure Mathematics examination from 2019. If there are any significant changes to the
specification Pearson will inform centres to let them know. Changes will also be posted on
our website.
For more information please visit qualifications.pearson.com
Using this specification
This specification has been designed to give guidance to teachers and encourage effective
delivery of the qualification. The following information will help you get the most out of the
content and guidance.
Compulsory content: arranged according to topic headings, as summarised in
Section 2: Further Mathematics content.
Examples: we have included examples to exemplify content statements to support teaching
and learning. It is important to note that these examples are for illustrative purposes only
and centres can use other examples. We have included examples that are easily understood
and recognised by international centres.
Qualification aims and objectives
The Pearson Edexcel International GCSE in Further Pure Mathematics qualification enables
students to:
•
study knowledge of mathematical techniques beyond International GCSE Mathematics
content
Pearson Edexcel International GCSE in Further Pure Mathematics –
Specification – Issue 1 – January 2016 © Pearson Education Limited 2016
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•
provide a course of study for those whose mathematical competence may have
developed early
•
develop an understanding of mathematical reasoning and processes, and the ability to
relate different areas of mathematics
•
enable students to acquire knowledge and skills with confidence, satisfaction and
enjoyment
•
develop mathematical skills for further study in the subject or related areas.
Why choose Edexcel qualifications?
Pearson – the world’s largest education company
Edexcel academic qualifications are from Pearson, the UK’s largest awarding organisation.
With over 3.4 million students studying our academic and vocational qualifications
worldwide, we offer internationally recognised qualifications to schools, colleges and
employers globally.
Pearson is recognised as the world’s largest education company, allowing us to drive
innovation and provide comprehensive support for Edexcel students to acquire the
knowledge and skills they need for progression in study, work and life.
A heritage you can trust
The background to Pearson becoming the UK’s largest awarding organisation began in 1836,
when a royal charter gave the University of London its first powers to conduct exams and
confer degrees on its students. With over 150 years of international education experience,
Edexcel qualifications have firm academic foundations, built on the traditions and rigour
associated with Britain’s educational system.
Results you can trust
Pearson’s leading online marking technology has been shown to produce exceptionally
reliable results, demonstrating that at every stage, Edexcel qualifications maintain the
highest standards.
Developed to Pearson’s world-class qualifications standards
Pearson’s world-class standards mean that all Edexcel qualifications are developed to be
rigorous, demanding, inclusive and empowering. We work collaboratively with a panel of
educational thought-leaders and assessment experts, to ensure that Edexcel qualifications
are globally relevant, represent world-class best practice and maintain a consistent
standard.
For more information on the World Class Qualification process and principles please go to
Appendix 2 or visit our website: uk.pearson.com/world-class-qualifications
Why choose Pearson Edexcel International GCSE in
Further Pure Mathematics?
We’ve listened to feedback from all parts of the International school and UK Independent
school subject community, including a large number of teachers. We’ve made changes that
will engage students and develop their skills to progress to further study of Mathematics and
a wide range of other subjects. Our content and assessment approach has been designed to
meet students’ needs and sits within our wider subject offer for Mathematics, providing the
next step on from both our International GCSE qualifications in Mathematics (Specification A
and B).
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Extended Mathematics course of study – We have designed this qualification to enhance
the learning for students whose mathematical competence may have developed early, or
who wish to pursue their interest in the subject. It provides an additional Level 2
International GCSE qualification, which extends mathematical techniques beyond those
covered in International GCSE/GCSE in Mathematics.
Clear and straightforward question papers – Our question papers are clear and
designed specifically to target the Higher Tier (grades 9–4 with grade 3 allowed) making
them accessible for students in this ability range. Our mark schemes are straightforward, so
that the assessment requirements are clear.
Broaden and deepen students’ skills – We have designed the International GCSE to
extend students’ knowledge by broadening and deepening skills, for example:
•
Students use numerical skills in both a purely mathematical way and in real-life
situations
•
Students use algebra and calculus to set up and solve problems
•
Students develop competence and confidence when manipulating mathematical
expressions
•
Students use vectors and rates of change to model situations.
Supports progression to A Level – Our qualifications enable successful progression onto
A Level and beyond. Through our world-class qualification development process we have
consulted with International A Level and GCE A Level teachers, as well as university
professors to validate the appropriacy of this qualification including the content, skills and
assessment structure. More information about all of our Mathematics qualifications can be
found on our Edexcel International GCSE pages at: qualifications.pearson.com
Supporting you in planning and implementing this
qualification
Planning
•
Our Getting Started Guide gives you an overview of the Pearson Edexcel International
GCSE in Further Pure Mathematics to help you understand the changes to content and
assessment, and to help you understand what these changes mean for you and your
students.
•
We will provide you with a course planner and editable schemes of work.
•
Our mapping documents highlight key differences between the new and 2009 legacy
qualification.
Teaching and learning
•
Our skills maps will highlight skills areas that are naturally developed through the study
of mathematics, showing connections between areas and opportunities for further
development.
•
Print and digital learning and teaching resources – promotes any time, any place learning
to improve student motivation and encourage new ways of working.
Preparing for exams
We will also provide a range of resources to help you prepare your students for the
assessments, including:
•
specimen papers to support formative assessments and mock exams
•
examiner commentaries following each examination series.
Pearson Edexcel International GCSE in Further Pure Mathematics –
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ResultsPlus
ResultsPlus provides the most detailed analysis available of your students’ exam
performance. It can help you identify the topics and skills where further learning would
benefit your students.
examWizard
A free online resource designed to support students and teachers with exam preparation and
assessment.
Training events
In addition to online training, we host a series of training events each year for teachers to
deepen their understanding of our qualifications.
Get help and support
Our subject advisor service will ensure you receive help and guidance from us. You can sign
up to receive the Edexcel newsletter to keep up to date with qualification updates and
product and service news.
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Qualification at a glance
The Pearson Edexcel International GCSE in Further Pure Mathematics comprises of two
externally assessed papers.
This specification is offered through a single tier.
Questions are targeted at grades in the range 9–4, with 3 allowed.
Students whose level of achievement is below the minimum judged by Pearson to be of
sufficient standard will receive an unclassified U result.
Paper overview
Paper 1
*Component/paper
code 4PM1/01
Paper 2
*Component/paper
code 4PM1/02
•
Externally assessed
•
Availability: January and June
•
First assessment: June 2019
Each paper is 50% of
the total International
GCSE
Content summary
•
Number
•
Algebra and calculus
•
Geometry and trigonometry
Assessment
•
Each paper is assessed through a 2-hour examination set and marked by Pearson.
•
The total number of marks for each paper is 100.
•
Each paper will consist of around 11 questions with varying mark allocations per
question, which will be stated on the paper.
•
Each paper will contain questions from any part of the specification content, and the
solution of any questions may require knowledge of more than one section of the
specification content.
•
The paper will have approximately 40% of the marks distributed evenly over grades
4 and 5 and approximately 60% of the marks distributed evenly over grades 6, 7, 8
and 9.
•
A formulae sheet (Appendix 4) will be included in the written examinations.
•
A calculator may be used in the examinations (see page 22 for more information).
* See Appendix 1 for a description of this code and all the other codes relevant to this
qualification.
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Pearson Edexcel International GCSE in Further Pure Mathematics –
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2 Further Pure Mathematics content
1: Logarithmic functions and indices
11
2: The quadratic function
12
3: Identities and inequalities
13
4: Graphs
14
5: Series
14
6: The binomial series
14
7: Scalar and vector quantities
15
8: Rectangular Cartesian coordinates
16
9: Calculus
17
10: Trigonometry
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Content
Externally assessed
Description
The Pearson Edexcel International GCSE in Further Pure Mathematics requires students to
demonstrate application and understanding of the following:
Number
•
Use numerical skills in a purely mathematical way and in real-life situations.
Algebra and calculus
•
Use algebra and calculus to set up and solve problems.
•
Develop competence and confidence when manipulating mathematical expressions.
•
Construct and use graphs in a range of situations.
Geometry and trigonometry
•
Understand the properties of shapes, angles and transformations.
•
Use vectors and rates of change to model situations.
•
Use coordinate geometry.
•
Use trigonometry.
Students will be expected to have a thorough knowledge of the content common to the
Pearson Edexcel International GCSE in Mathematics (Specification A) (Higher Tier) or
Pearson Edexcel International GCSE in Mathematics (Specification B).
Questions may be set which assumes knowledge of some topics covered in these
specifications, however knowledge of statistics and matrices will not be required.
Students will be expected to carry out arithmetic and algebraic manipulation, such as being
able to change the subject of a formula and evaluate numerically the value of any variable in
a formula, given the values of the other variables.
The use and notation of set theory will be adopted where appropriate.
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Assessment information
Each paper is assessed through a 2-hour examination set and marked by Pearson.
The total number of marks for each paper is 100.
Each paper will consist of around 11 questions with varying mark allocations per question,
which will be stated on the paper.
Each paper will contain questions from any part of the specification content, and the solution
of any questions may require knowledge of more than one section of the specification
content.
Each paper will have approximately 40% of the marks distributed evenly over grades
4 and 5 and approximately 60% of the marks distributed evenly over grades 6, 7, 8 and 9.
Diagrams will not necessarily be drawn to scale and measurements should not be taken from
diagrams unless instructions to this effect are given.
A formulae sheet (Appendix 4) will be included in the written examinations.
A calculator may be used in the examinations (please see page 22 for further information).
Questions will be set in SI units (international system of units).
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Logarithmic functions and indices
What students need to learn
x
Notes
A
The functions a and logb x (where
natural number greater than one)
b is a
B
Use and properties of indices and
logarithms, including change of base
A knowledge of the shape of the graphs
x
of a and logb x is expected, but not a
formal expression for the gradient.
To include:
log=
log a x + log a y ,
a xy
x
log
log a x − log a y ,
=
a
y
log a x k = k log a x,
log a a = 1
log a 1 = 0
The solution of equations of the form
a x = b.
Students may use the change of base
formulae:
C
Simple manipulation of surds
log a x =
log b x
log b a
log a b =
1
log b a
Students should understand what surds
represent and their use for exact
answers.
Manipulation will be very simple.
For example:
5 3+2 3 =
7 3
48 = 4 3
D Rationalising the denominator
10 ×
1
1
=
2 5 or
2− 3
5
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2
The quadratic function
What students need to learn
Notes
A
The manipulation of quadratic expressions
Students should be able to factorise
quadratic expressions and complete the
square.
B
The roots of a quadratic equation
Students should be able to use the
discriminant to identify whether the
roots are equal real, unequal real or not
real.
C
Simple examples involving functions of the
roots of a quadratic equation
Students are expected to understand
and use:
ax2 + bx + c = 0
has roots
α, β =
−b ± b2 − 4ac
2a
and forming an equation with given
roots, which are expressed in terms of
α and β :
=
α+β
12
−b
c
a
nd αβ
=
a
a
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Identities and inequalities
What students need to learn
Notes
A
Simple algebraic division
Division by (x + a), (x – a), (ax + b) or
(ax – b) will be required.
B
The factor and remainder theorems
Students should know that if f(x) = 0
when x = a, then (x – a) is a factor of
f(x).
Students may be required to factorise
cubic expressions such as:
x3 + 3x2 – 4 and 6x3 + 11x2 – x – 6, when
a factor has been provided.
Students should be familiar with the
terms ‘quotient’ and ‘remainder’ and be
able to determine the remainder when
the polynomial f(x) is divided by (ax + b)
or (ax – b).
C
Solutions of equations, extended to include
the simultaneous solution of one linear and
one quadratic equation in two variables
D Simple inequalities, linear and quadratic
The solution of a cubic equation
containing at least one rational root may
be set.
For example
ax + b > cx + d ,
2
px + qx + r < sx 2 + tx + u
E
The graphical representation of linear
inequalities in two variables
The emphasis will be on simple
questions designed to test fundamental
principles.
Simple problems on linear programming
may be set.
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4
Graphs
What students need to learn
Notes
A
Graphs of polynomials and rational
functions with linear denominators
The concept of asymptotes parallel to
the coordinate axes is expected.
B
The solution of equations and
transcendental functions by graphical
methods
Non-graphical iterative methods are not
required.
5
Series
What students need to learn
Notes
∑ notation
The ∑ notation may be employed
wherever its use seems desirable.
A
Use of the
B
Arithmetic and geometric series
Knowledge of the general term of an
arithmetic series is required.
Use of the sum to n terms of an
arithmetic series is required.
Knowledge of the general term of a
geometric series is required.
Use of the sum to n terms of a finite
geometric series is required.
Use of the sum to infinity of a
convergent geometric series, including
the use of r < 1 is required.
Proofs of the above are not required.
6
The binomial series
What students need to learn
A
Use of the binomial series
Notes
(1 + x)n
Use of the series when:
(i)
n is a positive integer
(ii)
n is rational and x < 1
The validity condition for (ii) is
expected.
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Pearson Edexcel International GCSE in Further Pure Mathematics –
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7 Scalar and vector quantities
What students need to learn
Notes
A
Knowledge of the fact that if
The addition and subtraction of coplanar
vectors and the multiplication of a vector by
a scalar
B
Components and resolved parts of a vector
C
Magnitude of a vector
D Position vector
E
Unit vector
F
Use of vectors to establish simple properties
of geometrical figures
α1a + β1b = α 2a + β 2b,
a and b are non-parallel vectors,
=
β 2 , is expected.
then α1 α=
2 and β1
where
Use of the vectors
expected.
i and j will be
  
AB =
OB − OA =b − a
The ‘simple properties’ will, in general,
involve collinearity, parallel lines and
concurrency.
Position vector of a point dividing the
line AB in the ratio m : n is expected.
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Rectangular Cartesian coordinates
What students need to learn
Notes
A
The distance d between two points
( x1 , y1 ) and ( x2 , y2 ) is given by
The distance between two points
d 2 = ( x1 − x2 ) 2 + ( y1 − y2 ) 2
B
The point dividing a line in a given ratio
The coordinates of the point dividing the
line joining ( x1 , y1 ) and ( x2 , y2 ) in the
ratio m : n are given by
 nx1 + mx2 ny1 + my2 
,


m+n 
 m+n
C
Gradient of a straight line joining two points
D The straight line and its equation
The y = mx + c and y − y1 = m( x − x1 )
forms of the equation of a straight line
are expected to be known.
The interpretation of ax + by = c as a
straight line is expected to be known.
E
16
The condition for two lines to be parallel or
to be perpendicular
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Calculus
What students need to learn
Notes
A
No formal proofs of the results for axn,
sin ax, cos ax and eax will be required.
Differentiation and integration of sums of
multiples of powers of x (excluding
integration of
1 sin ax, cos ax, e ax
),
x
B
Differentiation of a product, quotient and
simple cases of a function of a function
C
Applications to simple linear kinematics and
to determination of areas and volumes
Understanding how displacement,
velocity and acceleration are related
using calculus.
The volumes will be obtained only by
revolution about the coordinate axes.
D Stationary points and turning points
E
Maxima and minima
Maxima and minima problems may be
set in the context of a practical problem.
Justification of maxima and minima will
be expected.
F
The equations of tangents and normals to
the curve y = f(x)
G Application of calculus to rates of change
and connected rates of change
f(x) may be any function which the
students are expected to be able to
differentiate.
The emphasis will be on simple
examples to test principles.
A knowledge of dy
for small
≈
dy
dx
dx
dx is expected.
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Trigonometry
What students need to learn
Notes
A
The formulae:
Radian measure, including use for arc
length and area of sector
s = rθ and A =
1 2
rθ
2
for a circle are expected to be known.
B
The three basic trigonometric ratios of
angles of any magnitude (in degrees or
radians) and their graphs
C
Applications to simple problems in two or
three dimensions (including angles between
a line and a plane and between two planes)
D Use of the sine and cosine formulae
To include the exact values for sine,
cosine and tangent of 30°, 45°, 60° (and
the radian equivalents), and the use of
these to find the trigonometric ratios of
related values such as 120°, 300°
General proofs of the sine and cosine
formulae will not be required.
The cosine formula will be given but
other formulae are expected to be
known. The area of a triangle in the
form:
1
ab sin C is expected to be known.
2
cos2 θ + sin 2 θ =
1
E
The identity
F
Use of the identity
tan θ =
sin θ
cos θ
G The use of the basic addition formulae of
trigonometry
cos2 θ + sin 2 θ =
1 is expected to be
known.
This will be provided on the formula
sheet.
Formal proofs of sin(A + B), cos(A + B)
formulae will not be required.
Questions using the formulae for
sin(A + B), cos(A + B), tan(A + B) may be
set; the formulae will be on the formula
sheet, for example:
sin(A + B) = sinAcosB + cosAsinB
Long questions, explicitly involving
excessive manipulation, will not be set.
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What students need to learn
Notes
H Solution of simple trigonometric equations
for a given interval
Students should be able to solve
equations such as:
π
3
sin( x − ) =for 0 < x < 2 π ,
2
4
cos(3 x + 30=
°)
1
for − 90° < x < 90° ,
2
tan
=
2 x 1 for 90° < x < 270°,
6cos2 x° + sin x° − 5 = 0 for 0  x < 360,
π 1

sin 2  x +  =for − π  x < π
6 2

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Pearson Edexcel International GCSE in Further Pure Mathematics –
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3 Assessment information
Assessment requirements
Paper number
Level
Assessment information
Number of marks
allocated in the
paper
Paper 1
Higher
Assessed through a 2-hour
examination set and marked
by Pearson.
100
The paper is weighted at
50% of the qualification,
targeted at grades 9–4 with 3
allowed.
Paper 2
Higher
Assessed through a 2-hour
examination set and marked
by Pearson.
100
The paper is weighted at 50%
of the qualification, targeted
at grades 9–4 with 3 allowed.
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Calculators
Students will be expected to have access to a suitable electronic calculator for all
examination papers. The electronic calculator should have these functions as a minimum:
•
1
x
+, −, ×, ÷, π , x 2 , x , , x y , ln x, e , standard form, sine, cosine, tangent and their inverses in
x
degrees and decimals of a degree or radians.
Prohibitions
Calculators with any of the following facilities are prohibited in all examinations:
•
databanks
•
retrieval of text or formulae
•
QWERTY keyboards
•
built-in symbolic algebra manipulations
•
symbolic differentiation or integration.
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Assessment objectives and weightings
% in
International
GCSE
AO1
Demonstrate a confident knowledge of the techniques of
pure mathematics required in the specification
30–40%
AO2
Apply a knowledge of mathematics to the solutions of
problems for which an immediate method of solution is not
available and which may involve knowledge of more than
one topic in the specification
20–30%
Write clear and accurate mathematical solutions
35–50%
AO3
TOTAL
100%
Relationship of assessment objectives to units
Unit number
Assessment objective
AO1
AO2
AO3
Papers 1
15–20%
10–15%
17.5–25%
Papers 2
15–20%
10–15%
17.5–25%
Total for
International GCSE
30–40%
20–30%
35–50%
All components will be available for assessment from June 2019.
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4 Administration and general information
Entries
Details of how to enter students for the examinations for this qualification can be found in
our International Information Manual. A copy is made available to all examinations officers
and is available on our website.
Students should be advised that, if they take two qualifications in the same subject,
colleges, universities and employers are very likely to take the view that they have achieved
only one of the two GCSEs/International GCSEs. Students or their advisers who have any
doubts about subject combinations should check with the institution to which they wish to
progress before embarking on their programmes.
Access arrangements, reasonable adjustments, special consideration
and malpractice
Equality and fairness are central to our work. Our equality policy requires all students to
have equal opportunity to access our qualifications and assessments, and our qualifications
to be awarded in a way that is fair to every student.
We are committed to making sure that:
•
students with a protected characteristic (as defined by the UK Equality Act 2010) are not,
when they are undertaking one of our qualifications, disadvantaged in comparison to
students who do not share that characteristic
•
all students achieve the recognition they deserve for undertaking a qualification and that
this achievement can be compared fairly to the achievement of their peers.
Language of assessment
Assessment of this qualification will only be available in English. All student work must be in
English.
We recommend that students are able to read and write in English at Level B2 of the
Common European Framework of Reference for Languages.
Access arrangements
Access arrangements are agreed before an assessment. They allow students with special
educational needs, disabilities or temporary injuries to:
•
access the assessment
•
show what they know and can do without changing the demands of the assessment.
The intention behind an access arrangement is to meet the particular needs of an individual
student with a disability without affecting the integrity of the assessment. Access
arrangements are the principal way in which awarding bodies comply with the duty under
the Equality Act 2010 to make ‘reasonable adjustments’.
Access arrangements should always be processed at the start of the course. Students will
then know what is available and have the access arrangement(s) in place for assessment.
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Reasonable adjustments
The UK Equality Act 2010 requires an awarding organisation to make reasonable
adjustments where a student with a disability would be at a substantial disadvantage in
undertaking an assessment. The awarding organisation is required to take reasonable steps
to overcome that disadvantage.
A reasonable adjustment for a particular student may be unique to that individual and
therefore might not be in the list of available access arrangements.
Whether an adjustment will be considered reasonable will depend on a number of factors,
including:
•
the needs of the student with the disability
•
the effectiveness of the adjustment
•
the cost of the adjustment; and
•
the likely impact of the adjustment on the student with the disability and other students.
An adjustment will not be approved if it involves unreasonable costs to the awarding
organisation, timeframes or affects the security or integrity of the assessment. This is
because the adjustment is not ‘reasonable’.
Special consideration
Special consideration is a post-examination adjustment to a student's mark or grade to
reflect temporary injury, illness or other indisposition at the time of the examination/
assessment, which has had, or is reasonably likely to have had, a material effect on a
candidate’s ability to take an assessment or demonstrate their level of attainment in an
assessment.
Further information
Please see our website for further information about how to apply for access arrangements
and special consideration.
For further information about access arrangements, reasonable adjustments and special
consideration please refer to the JCQ website: www.jcq.org.uk
Malpractice
Candidate malpractice
Candidate malpractice refers to any act by a candidate that compromises or seeks to
compromise the process of assessment or that undermines the integrity of the qualifications
or the validity of results/certificates.
Candidate malpractice in examinations must be reported to Pearson using a JCQ Form M1
(available at www.jcq.org.uk/exams-office/malpractice). The form can be emailed to
pqsmalpractice@pearson.com or posted to: Investigations Team, Pearson,
190 High Holborn, London, WC1V 7BH. Please provide as much information and supporting
documentation as possible. Note that the final decision regarding appropriate sanctions lies
with Pearson.
Failure to report malpractice constitutes staff or centre malpractice.
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Staff/centre malpractice
Staff and centre malpractice includes both deliberate malpractice and maladministration of
our qualifications. As with candidate malpractice, staff and centre malpractice is any act that
compromises or seeks to compromise the process of assessment or that undermines the
integrity of the qualifications or the validity of results/certificates.
All cases of suspected staff malpractice and maladministration must be reported
immediately, before any investigation is undertaken by the centre, to Pearson on
a JCQ Form M2(a) (available at www.jcq.org.uk/exams-office/malpractice).
The form, supporting documentation and as much information as possible can be emailed to
pqsmalpractice@pearson.com or posted to: Investigations Team, Pearson,
190 High Holborn, London, WC1V 7BH. Note that the final decision regarding appropriate
sanctions lies with Pearson.
Failure to report malpractice itself constitutes malpractice.
More-detailed guidance on malpractice can be found in the latest version of the document
JCQ General and vocational qualifications Suspected Malpractice in Examinations and
Assessments, available at www.jcq.org.uk/exams-office/malpractice
Awarding and reporting
The International GCSE qualification will be graded and certificated on a six-grade scale from
9 to 4, with 3 allowed using the total subject mark where 9 is the highest grade. Individual
components are not graded. The first certification opportunity for the Pearson Edexcel
International GCSE in Further Pure Mathematics will be in June 2019. Students whose level
of achievement is below the minimum judged by Pearson to be of sufficient standard to be
recorded on a certificate will receive an unclassified U result.
Student recruitment and progression
Pearson’s policy concerning recruitment to our qualifications is that:
•
they must be available to anyone who is capable of reaching the required standard
•
they must be free from barriers that restrict access and progression
•
equal opportunities exist for all students.
Prior learning and other requirements
Students will be expected to have a thorough knowledge of the content common to the
Pearson Edexcel International GCSE in Mathematics (Specification A) (Higher Tier) or
Pearson Edexcel International GCSE in Mathematics (Specification B).
Progression
Students can progress from this qualification to:
•
the GCE Advanced Subsidiary (AS) and Advanced Level in Mathematics, Further
Mathematics and Pure Mathematics
•
the International Advanced Subsidiary (AS) and Advanced Level in Mathematics, Further
Mathematics and Pure Mathematics
•
other equivalent, Level 3 Mathematics qualifications
•
further study in other areas where mathematics is required
•
other further training or employment where numerate skills and knowledge are required.
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Appendices
Appendix 1: Codes
31
Appendix 2: Pearson World Class Qualification Design Principles
33
Appendix 3: Transferable skills
35
Appendix 4: Formulae sheet for examinations
37
Appendix 5: Formulae to learn
39
Appendix 6: Notation
43
Appendix 7: Glossary
45
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Appendix 1: Codes
Type of code
Use of code
Code
Subject codes
The subject code is used by centres to
enter students for a qualification.
Pearson Edexcel
International GCSE
Further Pure
Mathematics: 4PM1
Paper codes
These codes are provided for
information. Students need to be
entered for individual papers.
Paper 1: 4PM1/01
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Paper 2: 4PM1/02
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Appendix 2: Pearson World Class Qualification
Design Principles
Pearson’s world-class qualification design principles mean that all Edexcel qualifications are
developed to be rigorous, demanding, inclusive and empowering.
We work collaboratively to gain approval from an external panel of educational thoughtleaders and assessment experts from across the globe. This is to ensure that Edexcel
qualifications are globally relevant, represent world-class best practice in qualification and
assessment design, maintain a consistent standard and support learner progression in
today’s fast changing world.
Pearson’s Expert Panel for World-class Qualifications is chaired by Sir Michael Barber, a
leading authority on education systems and reform. He is joined by a wide range of key
influencers with expertise in education and employability.
“I’m excited to be in a position to work with the global leaders in curriculum and assessment
to take a fresh look at what young people need to know and be able to do in the 21st
century, and to consider how we can give them the opportunity to access that sort of
education.” Sir Michael Barber.
Endorsement from Pearson’s Expert Panel for World-class Qualifications for
International GCSE development processes
“We were chosen, either because of our expertise in the UK education system, or because of
our experience in reforming qualifications in other systems around the world as diverse as
Singapore, Hong Kong, Australia and a number of countries across Europe.
We have guided Pearson through what we judge to be a rigorous world-class qualification
development process that has included:
•
Extensive international comparability of subject content against the highest-performing
jurisdictions in the world
•
Benchmarking assessments against UK and overseas providers to ensure that they are at
the right level of demand
•
Establishing External Subject Advisory Groups, drawing on independent subject-specific
expertise to challenge and validate our qualifications
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Importantly, we have worked to ensure that the content and learning is future oriented, and
that the design has been guided by Pearson’s Efficacy Framework. This is a structured,
evidence-based process which means that learner outcomes have been at the heart of this
development throughout.
We understand that ultimately it is excellent teaching that is the key factor to a learner’s
success in education but as a result of our work as a panel we are confident that we have
supported the development of Edexcel International GCSE qualifications that are outstanding
for their coherence, thoroughness and attention to detail and can be regarded as
representing world-class best practice.”
Sir Michael Barber (Chair)
Professor Sing Kong Lee
Chief Education Advisor, Pearson plc
Professor, National Institute of Education in
Singapore
Dr Peter Hill
Bahram Bekhradnia
Former Chief Executive ACARA
President, Higher Education Policy Institute
Professor Jonathan Osborne
Dame Sally Coates
Stanford University
Director of Academies (South), United
Learning Trust
Professor Dr Ursula Renold
Professor Bob Schwartz
Federal Institute of Technology,
Switzerland
Harvard Graduate School of Education
Professor Janice Kay
Jane Beine
Provost, University of Exeter
Head of Partner Development, John Lewis
Partnership
Jason Holt
CEO, Holts Group
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Appendix 3: Transferable skills
The need for transferable skills
In recent years, higher education institutions and employers have consistently flagged the
need for students to develop a range of transferable skills to enable them to respond with
confidence to the demands of undergraduate study and the world of work.
The Organisation for Economic Co-operation and Development (OECD) defines skills, or
competencies, as ‘the bundle of knowledge, attributes and capacities that can be learned
and that enable individuals to successfully and consistently perform an activity or task and
can be built upon and extended through learning.’[1]
To support the design of our qualifications, the Pearson Research Team selected and
evaluated seven global 21st-century skills frameworks. Following on from this process, we
identified the National Research Council’s (NRC) framework [ 2] as the most evidence-based
and robust skills framework, and have used this as a basis for our adapted skills framework.
The framework includes cognitive, intrapersonal skills and interpersonal skills.
The skills have been interpreted for this specification to ensure they are appropriate for the
subject. All of the skills listed are evident or accessible in the teaching, learning and/or
assessment of the qualification. Some skills are directly assessed. Pearson materials will
support you in identifying these skills and developing these skills in students.
The table overleaf sets out the framework and gives an indication of the skills that can be
found in the Pearson Edexcel International GCSE in Further Pure Mathematics and indicates
the interpretation of the skill in this area. A full subject interpretation of each skill, with
mapping to show opportunities for students’ development is provided on the subject pages
of our website: qualifications.pearson.com
1
OECD (2012), Better Skills, Better Jobs, Better Lives (2012):
http://skills.oecd.org/documents/OECDSkillsStrategyFINALENG.pdf
2
Koenig, J. A. (2011) Assessing 21st Century Skills: Summary of a Workshop, National Research Council
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Cognitive processes
and strategies:
• Critical thinking
• Problem solving
Cognitive skills
• Analysis
• Reasoning
• Interpretation
• Decision making
• Adaptive learning
• Executive function
Creativity:
Problem solving for
translating problems in
mathematical or
non-mathematical
contexts into a process
or a series of
mathematical processes
and solving them.
• Creativity
• Innovation
Intellectual
openness:
• Adaptability
• Personal and social
responsibility
• Continuous learning
Intrapersonal skills
• Intellectual interest
and curiosity
Work ethic/
conscientiousness:
• Initiative
• Self-direction
• Responsibility
• Perseverance
• Productivity
• Self-regulation
(metacognition,
forethought,
reflection)
• Ethics
• Integrity
Interpersonal skills
Initiative for using
mathematical
knowledge,
independently
(without guided
learning), to further
own understanding.
Positive core self
evaluation:
• Self-monitoring/selfevaluation/selfreinforcement
Teamwork and
collaboration:
• Communication
• Collaboration
Communication to
communicate a
mathematical process
or technique
(verbally or written) to
peers and teachers and
answer questions from
others.
• Teamwork
• Co-operation
• Interpersonal skills
Leadership:
• Leadership
• Responsibility
• Assertive
communication
• Self-presentation
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Appendix 4: Formulae sheet for examinations
Mensuration
Surface area of sphere = 4π r 2
Curved surface area of cone = π r × slant height
4
Volume of sphere = π r 3
3
Series
Arithmetic series
Sum to n terms, Sn =
Geometric series
n
[ 2a + (n − 1)d ]
2
Sum to n terms, Sn =
Sum to infinity,
=
S∞
Binomial series
(1 + x )
n
=1 + nx +
Calculus
a (1 − r n )
(1 − r )
a
1− r
r <1
n(n − 1) 2
n(n − 1)...(n − r + 1) r
x + ... +
x + ...
2!
r!
for
x < 1, n ∈ 
Quotient rule (differentiation)
d  f ( x)  f ′( x)g( x) − f ( x)g′( x)

=
2
dx  g( x) 
[g( x)]
Trigonometry
Cosine rule
In triangle ABC: a 2 = b 2 + c 2 − 2bc cos A
tan θ =
sin θ
cos θ
sin( A
=
+ B ) sin A cos B + cos A sin B
sin ( A=
− B ) sin A cos B − cos A sin B
cos ( A
=
+ B ) cos A cos B − sin A sin B
cos ( =
A − B ) cos A cos B + sin A sin B
tan A + tan B
tan ( A + B ) =
1 − tan A tan B
tan A − tan B
tan ( A − B ) =
1 + tan A tan B
Logarithms
log a x =
logb x
logb a
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Appendix 5: Formulae to learn
This appendix gives formulae that students are expected to remember and will not be
included on the formula sheet provided in the examination papers.
Logarithmic functions and indices
log=
log a x + log a y
a xy
x
=
log
log a x − log a y
a
y
log a x k = k log a x
1
= − log a x
x
log a a = 1
log a
log a 1 = 0
log a b =
1
log b a
Quadratic equations
ax 2 + bx + c =
0 has roots x =
−b ± b2 − 4ac
2a
b
c
ax 2 + bx + c =
0 are α and β then α + β =
− and αβ =
a
a
2
and the equation can be written x − (α + β ) x + αβ =
0
when the roots of
Series
Arithmetic series
nth term = l = a + ( n −1) d
Geometric series
nth term = ar n −1
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Coordinate geometry
The gradient of the line joining two points ( x1 , y1 ) and ( x2 , y2 ) is given by
The distance
y2 − y1
x2 − x1
d between two points ( x1 , y1 ) and ( x2 , y2 ) is given by d 2 = ( x1 − x2 ) 2 + ( y1 − y2 ) 2
The coordinates of the point dividing the line joining ( x1 , y1 ) and
( x2 , y2 ) in the ratio m : n are
 nx1 + mx2 ny1 + my2 
,


m+n 
 m+n
Calculus
Differentiation:
Function
n
Derivative
x
nxn – 1
sin ax
acos ax
cos ax
–asin ax
ax
e
aeax
f ( x )g( x )
f ′( x )g( x ) + f ( x )g′( x )
f (g( x ))
f ′(g( x ))g′( x )
Integration:
Function
Integral
xn
1 n +1
x + c n ≠ −1
n +1
sin ax
1
− cos ax + c
a
cos ax
1
sin ax + c
a
eax
1 ax
e +c
a
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Area and volume:
Area between a curve and the x-axis =
b
∫ y dx, y ≥ 0
a
b
∫ y dx , y < 0
a
Area between a curve and the y-axis =
d
∫ x dy, x ≥ 0
c
d
∫ x dy , x < 0
c
Area between
g(x) and f(x) =
Volume of revolution =
∫
b
a
∫
b
a
g( x ) − f ( x ) dx
π y 2 dx
or
∫
d
c
π x 2dy
Trigonometry
Radian measure:
length of arc
= rθ
area of sector
In triangle
ABC:
=
1 2
rθ
2
a
b
c
= =
sin A sin B sin C
cos2 θ + sin 2 θ =
1
area of triangle =
1
ab sin C
2
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Appendix 6: Notation
Notation used in the examination include the following:
{
}
the set of
n(A)
the number of elements in the set
{x: }
the set of all
∈
is an element of
∉
is not an element of
∅
the empty (null) set
E
the universal set
∪
union
∩
intersection
⊂
is a subset of
A′
the complement of the set

the set of natural numbers,

the set of integers numbers,
A
x such that
A
{1, 2, 3, …}
{0, ±1, ±2, ±3…}

p
: p ∈ , q ∈ + 
q


the set of rational numbers, 

the set of real numbers
x
x
the modulus of
≈
is approximately equal to
n
∑a
a1 + a2 +  + an
n
r 
 
n ( n − 1)( n − r + 1)
for n ∈ 
r!
ln x
the natural logarithm of
lg x
logarithm of
f ′ ( x)
the first derivative of
f: A → B
is a function under which each element of set
f: x  y
f is a function under which x is mapped to y
f(x)
the image of
i =1
f
-1
fg
i
x
x to base 10
f(x) with respect to x
x under the function f
the inverse relation of the function
the function
A has an image in set B
f
g followed by function f, i.e. f(g(x))
open interval on the number line
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closed interval on the number line
a

AB
the vector
a
the vector represented in magnitude and direction by
the vector from point
a
44

AB
A to point B
the magnitude of vector
a
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Appendix 7: Glossary
Term
Definition
Assessment
objectives
The requirements that students need to meet to succeed in the
qualification. Each assessment objective has a unique focus which
is then targeted in examinations or coursework. Assessment
objectives may be assessed individually or in combination.
External
assessment
An examination that is held at the same time and place in a global
region.
JCQ
Joint Council for Qualifications. This is a group of UK exam boards
that develop policy related to the administration of examinations.
Linear
Qualifications that are linear have all assessments at the end of a
course of study. It is not possible to take one assessment earlier in
the course of study.
Unit
A modular qualification will be divided into a number of units. Each
unit will have its own assessment.
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